MATHCOUNTS
1999 State Competition
Countdown Round
®
How many of the first 100 positive
integers are neither perfect squares
nor perfect cubes?
Answer: 88 (integers)
A palindrome is a number that
reads the same forwards and
backwards. How many
palindromes are between 1000
and 10,000?
Answer: 90 (palindromes)
The surface of a 4” x 5” x 6”
block is painted yellow. If the
block is separated into one-inch
cubes, what is the number of
cubes with exactly one yellow
face?
Answer: 52 (cubes)
The lengths of two sides of a
right triangle are 2 11 meters
and 6 3 meters. The number of
meters in the length of the third
side is a whole number. How
many meters are in the third side
of the triangle?
Answer: 8 (meters)
A bag contains yellow and blue
marbles. The probability of
selecting a yellow marble from
8
the bag is 17 . If there are 27 blue
marbles in the bag, what is the
number of yellow marbles?
Answer: 24 (marbles)
A uniform piece of lumber is
12 ft long and weighs 30 lb. If a
5 ft length is removed, what is
the number of pounds in the
remaining piece? Express your
answer as a mixed number.
Answer: 17
1
2
(pounds)
If q, r and s each represent a
different integer 0-9, what is the
value of s?
qqqq
rrrr
+ ssss
rqqqs
Answer: 8
Compute: (26  24 ) .
2
2 2
Answer: 10,000
1
9
What number is 11 %
of 13 239?
Answer: 1471
For what whole number n is
n
n 1
10  695000  10 ?
Answer: 5
What is the ratio of 0.16 to 0.83?
Express your answer as a
common fraction.
Answer:
1
5
The measure of the angles of a
pentagon are in the ratio of
3:3:3:4:5. What is the number of
degrees in the measure of the
largest angle?
Answer: 150 (degrees)
A digital clock shows hours and
minutes. How many times
during a 12-hour period will the
sum of the digits on this clock
be 3?
Answer: 12 (times)
It takes 6 days for a team of
6 workers to carpet a house. At
the same rate, what is the
minimum number of additional
workers needed to carpet the
same house in just 5 days?
Answer: 2 (workers)
If a  b  12 and a  b  6 , what
2
2
is the value of a  2a  2b  b ?
Answer: 114
John has some loaves of bread in
a bag. If he had 6 more loaves,
he would have three times as
many. How many loaves does
John have in the bag?
Answer: 3 (loaves)
If a two-digit prime number is
selected at random, what is the
probability that the sum of its
digits is 9?
Answer: 0
In right triangle ABC, CAB is a
right angle. Point M is the midpoint
of BC. What is the number of
centimeters in the length of median
AM? Express your answer as a
decimal to the nearest tenth.
Answer: 2.5 (centimeters)
What is the number of degrees in
the exterior angle of the largest
angle of a triangle if the interior
angles are in the ratio of 2:3:5?
Answer: 90 (degrees)
A 12 feet by 8 feet rectangular
Persian rug is centered within a
rectangular room so that a region
of constant width surrounds the
carpet. Given that 96 square feet
of flooring is not covered by the
rug, what is the number of feet in
the measure of the longer
dimension of the room?
Answer: 16 (feet)
What is the least positive integer
that is not a factor of 7! ?
Answer: 11
A right circular cone is inscribed in a
right circular cylinder. The volume
of the cylinder is 72p cubic
centimeters. What is the number of
cubic centimeters in the space inside
the cylinder but outside
the cone? Express your
answer in terms of p.
Answer: 48p (cubic centimeters)
Compute
1
12 2
% of 240.
Answer: 30
In an isosceles trapezoid, the
lengths of the bases are 8 cm and
16 cm. The length of a diagonal
is 13 cm. What is the number of
square centimeters in the area of
the trapezoid?
Answer: 60 (square centimeters)
What is the greatest two-digit
whole number that can be
expressed as the product of two
consecutive whole numbers?
Answer: 90
What is the greatest three-digit
number that is divisible by 6?
Answer: 996
If the measure of each side of a
square is decreased by 10%, by
what percent is the area
decreased?
Answer: 19 (percent)
What is the units digit of
45
89
23  67 ?
Answer: 0
All squares have the same value,
and all circles have the same
value. What is the value of the
sum of three circles?
+
+
+
+
= 21
+
+
+
+
= 19
+
+
=?
Answer: 9
Six less than a number is
multiplied by 6. Eight less than
the same number is multiplied
by 8. The two results are equal.
What is the number?
Answer: 14
When Sarah turns 36 years old,
her father will be 62. Currently,
1
Sarah is 3 of her father’s age.
How many years old is Sarah
now?
Answer: 13 (years)
The sum of seven consecutive
even numbers is 13,902. What is
the smallest of the seven
numbers?
Answer: 1980
Eleven coins total $1.19, but no
combination of the coins will
give a value of exactly $1. How
many of the coins are quarters?
Answer: 3 (quarters)
Jimi is taking a bicycle trip and
needs to arrive by 6 o’clock. If he
travels at an average rate of 15 mph,
he’ll arrive at his destination one
hour early, but if he travels at an
average rate of 10 mph, he’ll arrive
one hour late. How many miles
away is his destination?
Answer: 60 (miles)
Mark baked cookies for the
neighborhood children. He gave
each child 6 cookies, and he had
7 cookies remaining. He was
going to give another cookie to
each child, but he was 1 cookie
short. How many cookies had
Mark baked?
Answer: 55 (cookies)
An eight-digit number contains
two 5’s, two 6’s, two 7’s and two 8’s.
The 5’s are separated by one digit,
the 6’s are separated by two digits,
the 7’s are separated by three digits,
and the 8’s are separated by four
digits. What is the greatest possible
value for the number?
Answer: 85,756,876
3
5
Calculate: (243) .
Answer: 27
Calculate and express your
answer as a common fraction:
2
2
3
 
2
5
2
7
105
Answer:
71
What is the least positive integer
n such that 16n  n ?
Answer: 5
How many different
three-member committees
can be chosen from a
group of ten people?
Answer: 120 (committees)
A rectangle with length 8 units
and width 5 units is centered at
the origin of a coordinate plane,
and its longer side is parallel to
the x-axis. What percent of the
rectangular region is to the right
of the line with equation x  2 ?
Answer: 75 (percent)
What is the smallest natural
number that has exactly five
factors?
Answer: 16
For what value of n does
n5
1
5  125
?
Answer: 2
What is the sum of the real values
of x for which the following
expression is undefined? Express
your answer as a common
fraction.
8
5
2
x  3
Answer:
15
2
A 3” x 5” photograph can be
enlarged to a panoramic view that
is 5” x 10”. What is the ratio of
the area of the original
photograph to the area of the
enlargement? Express your
answer as a common fraction.
Answer:
3
10
The sum of two numbers is 9,
and their difference is 7. What is
the positive difference of the
squares of these two numbers?
Answer: 63
One diagonal of a rhombus
measures 48 inches. Given that
one side of the rhombus has
length 26 inches, what is the
number of square inches in the
area of the rhombus?
Answer: 480 (square inches)
A centipede climbs a 40-foot tree.
Each day he climbs 5 feet, and
each night he slides down 2 feet.
In how many days will the
centipede reach the top of the
tree?
Answer: 13 (days)
For how many integers n
is |n - 5| < 3 ?
Answer: 5 (integers)
Bill, Frank and Grete are playing a
game. Bill is three times as likely to
win the game as Frank, and Frank is
two times as likely to win the game
as Grete. Assuming no ties, what is
the probability that Grete will win
the game? Express your answer as a
common fraction.
Answer:
1
9
What is the greatest rational
number n such that
2
n  11n  24  0 ?
Answer: 8
What is the sum of the
reciprocals of 1, 2, 3, 4 and 5?
Express your answer as a
common fraction.
137
Answer:
60
A jar contains 60 marbles, each
colored either green or yellow, all
the same size. There are twice as
many green marbles as yellow in
the jar. How many yellow
marbles must be added to the jar
so the probability of choosing a
1
green marble is 2 ?
Answer: 20 (marbles)
Each of the 30 students in a
physical education class plays the
same number of minutes of fullcourt basketball in a class period.
Ten students play at a time, and
the class period lasts 51 minutes.
What is the number of minutes
each student plays?
Answer: 17 (minutes)
A space diagonal of a given cube
is 10 3 inches long. What is the
number of square inches in the
surface area of the cube?
Answer: 600 (square inches)
A monument is constructed so
that the floor is a regular hexagon
with sides measuring 8 feet.
What is the number of square feet
in the area of the floor? Express
your answer in simplest radical
form.
Answer: 96 3 (square feet)
In China, small spheres are used
to exercise a person’s hands. If
the diameter of one of these
spheres measures 2 inches, how
many square inches are in its
surface area? Express your
answer in terms of p.
Answer: 4p (square inches)
For what value of n does
n
44444  2 ?
Answer: 10
Sam and Janet each have a whole
1
number of dollars, and 3 of
1
Sam’s money equals of Janet’s
2
money. The sum of their money
is more than $10. What is the
least whole number of dollars
they could have combined?
Answer: 15 (dollars)
What is the greatest digit that can
be placed in the box so that the
six-digit number shown is
divisible by 4?
352,9 2
Answer: 9
When you roll three number
cubes, each numbered 1 through
6, what is the probability that the
tops of the three cubes will show
either all even numbers or all odd
numbers? Express your answer
as a common fraction.
Answer:
1
4
For every 3 herks, there are 4
merks. Together, there are 84
herks and merks. How many
more merks than herks are there?
Answer: 12 (merks)
Given that 0.05n  1.25 ,
2
evaluate n  10 . Express your
answer as a decimal to the
nearest hundredth.
Answer: 0.25
Mark earns $8.00 per hour for the first
8 hours he works each day. If he works
more than 8 hours, he gets paid 1.5 times
his hourly rate for the extra hours. On
Monday, Mark worked from 7:30 AM to
6:30 PM and took an hour for lunch. If
his lunch hour is unpaid, how many
dollars did Mark earn on Monday?
Answer: 88 (dollars)
If three values from the list below
are selected at random without
replacement, what is the
probability that they are equal?
Express your answer as a
common fraction.
24
60
40% of 1
2
5
0.4
1
4
Answer:
2
5
Points A, B, C, D and E are five
points on a line segment with
endpoints A and E. The points are in
the order listed1 above from1left to
right. CD
= 2 AB, BC = 2 CD,
1
AB = 2 AE, and AE = 12 cm. What
is the number of centimeters in the
length of AD? Express your answer
as a decimal to the nearest tenth.
Answer: 10.5 (centimeters)
What is the maximum possible
value of x in the inequality
1
. ?
2 |2 x  5|  55
Answer: 8
Three straight lines intersect at O,
and OD bisects COE. The ratio
of COB to BOF is 7:2.
What is the number of degrees in
m COD?
Answer: 20 (degrees)
Jena starts on a trip at 8:00 AM
and travels at a rate of 50 mph.
Harvey starts out at 10:30 AM
the same day and travels the
same route at a rate of 75 mph.
What is the sum of the total miles
traveled by Harvey and Jena
when Harvey catches up with
Jena?
Answer: 750 (miles)
A right circular cylinder with
radius 2 inches and height 1 inch
has the same volume as a second
right circular cylinder. The
radius of the second cylinder is 1
inch. What is the number of
inches in the second cylinder’s
height?
Answer: 4 (inches)
For how many values of x does
|3x  2|  4  0 ?
Answer: 2 (values)
Using only quarters, dimes,
nickels, and pennies, what is the
minimum number of coins
needed to get 91 cents?
Answer: 6 (coins)
The measure of an exterior angle
o
of a triangle is 105 . If one of the
non-adjacent interior angles
o
measures 38 , what is the number
of degrees in the other
non-adjacent interior angle?
Answer: 67 (degrees)
Given that 36 feet of fencing are
needed to fence the sides of a
triangular region with sides in the
ratio 3:4:5, how many square feet
are in the area of the triangular
plot?
Answer: 54 (square feet)
Larry traveled from his house to
the mall at 40 mph and returned
along the same route at 60 mph.
What was his average speed in
miles per hour for the entire trip?
Answer: 48 (miles per hour)
How many seconds does it take
the second hand of a clock to
o
rotate through an angle of 36 ?
Answer: 6 (seconds)